nLab Sandbox

CQTS-QuantumIndustryDay-Nov2024.jpg?

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• Niccolò Cribiori, Dieter Lust: String dualities and modular symmetries in supergravity: a review, in: Half a century of Supergravity, Cambridge University Press [arXiv:2411.06516]

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A notorious bug:

The code

“[[Set]] [[topos]]”

renders as:

Settopos

(lacking the whitespace).

Similarly the code

“[[Set]] $topos$”

renders without the whitespace:

Set$topos$

This issue goes away when we are not at the beginning of a line. For instance

“A [[Set]] [[topos]]”

renders correctly as

A Set topos

and

“A [[Set]] $topos$”

renders correctly as

A Set $topos$

More testing:

topostopos topos topos

Settopos (two whitespaces)

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just another test

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The lattice of subtoposes

Since the maps in $\Delta_+$ are all strictly monotone, any object $[n]$ receives only (finitely many) morphisms from objects $[m]$ with $m\leq n$ whence all slices $\Delta_+/[n]$ are finite. This is sufficient for all Grothendieck topologies on $\Delta_+$ to be rigid whence all subtoposes of $Set^{\Delta_+^{op}}$ are essential and of presheaf type and their lattice is isomorphic to the lattice of Cauchy-complete full subcategories of $\Delta_+$.

This situation is familiar from $\Delta$ but in the latter case there are considerable fewer Cauchy-complete subcategories available, since an object $[n]$ having non-trivial idempotents its inclusion automatically requires the presence of all $[m]$ with $m\,$<$\,n$ in the subcategory whence there a countably many subtoposes corresponding to the $n$-truncated subcategories on the objects $[0],\dots [n]$.

For further details on the topologies, closure operators and sheaves involved in both cases see Rosset-Hansen-Endrullis).

Sugimoto string theory

The Sugimoto string theory is a non-supersymmetric version of type I string theory, given by 10d type IIB string theory on an $O9^+$ orientifold (instead of $O9^-$), hence with RR-field tadpole cancellation by 32 anti D9-branes (instead of plain D9-branes), whose gauge group is the symplectic group $USp(32)$.

References

The original article:

• Shigeki Sugimoto: Anomaly Cancellations in the Type I D9-anti-D9 System and the $USp(32)$ String Theory, Prog. Theor. Phys. 102 (1999) 685-699 [arXiv:hep-th/9905159, doi:10.1143/PTP.102.685]

Further discussion:

• Sanefumi Moriyama: $USp(32)$ String as Spontaneously Supersymmetry Broken Theory, Phys. Lett. B 522 (2001) 177-180 [arXiv:hep-th/0107203, doi:10.1016/S0370-2693(01)01278-3]

• Hector Parra de Freitas, p. 166 of: String Compactifications with Half-maximal Supersymmetry, PhD thesis, Université Paris-Saclay (2023) [hal/tel-04234221, pdf]